Update: In a new twist, Best Buy's board acknowledged late Tuesday that Dunn's personal conduct was being investigated. “Certain issues were brought to the board’s attention regarding Mr. Dunn’s personal conduct, unrelated to the company’s operations or financial controls, and an audit committee investigation was initiated,”Claire Koeneman of H+K Strategies, a spokeswoman for the Best Buy board of directors, told the Minneapolis Star Tribune. We'll post more about this news as we know more.

But as for the issue of his replacement...Best Buy’s new leader will need to possess a rare combination of qualities to stop the aging electronics retailer from going the way of Circuit City. Obviously, the company needs someone to revamp its much bemoaned customer service problem. But traditional retail experience alone won’t cut it—the company needs a visionary leader who can forge an entirely new path.

“One of the key questions is ‘What’s our strategy going to be? Downsize stores? Shift from stores to digital? Or is it to invent a concept in the U.S.?'” says Jason Goldberg, vice president of strategy and consumer experience at retail consultancy CrossView Inc. “The riskiest is to invent a new model in the U.S., but I really think that’s going to be the play for Best Buy. So that’s going to take a non-traditional CEO.”

Forbes reached out this morning to the nation’s top executive search companies, and zero of them got right back to us--probably because they're busy putting together pitches explaining why they’re the best people to find Dunn’s replacement. Here’s what the experts we spoke with say should be on the short list of qualities for the new CEO:

Know the customer

Best Buy’s biggest problem, hands-down, has come from not understanding its consumers. Akshay Rao, a professor of marketing at the University of Minnesota’s Carlson School of Management, has been predicting Dunn’s departure for some time now, based in part on what he calls “the China kerfuffle”—the company’s failed experiment in expanding into China. “That was a bad idea, because they hadn’t made the effort to be intimately knowledgeable about consumer behavior,” Rao says.

The search committee might want to look to former execs from companies known for great customer service grounded in understanding their customers—places like Southwest Airlines, Starbucks, or Nordstrom’s, says Mike Frommelt, principal and co-founder at Keystone Search, a boutique search firm based in Minneapolis, near Best Buy’s headquarters.

Another option, points out Goldberg, is choosing someone from a place like Williams-Sonoma, a company that has thrived in both its brick-and-mortar and e-commerce sites, precisely because it makes the shopping experience so enjoyable.

Know the brand

Best Buy sometimes seems to behave as though Amazon is an impossible competitor to beat, but that’s in fact a failure of imagination--and marketing, argues Rao. “Think about Chiquita bananas. A banana is a banana is a banana—unless it’s a Chiquita,” he says.

To survive, the electronics retailer needs to deeply understand the strengths and weaknesses of the Best Buy brand, and then leverage those to connect emotionally with consumers, Rao says. So Best Buy’s new CEO must either understand marketing or surround him or herself with the most creative minds out there.

One potential candidate from inside Best Buy, notes Goldberg, is Barry Judge, the chief marketing officer, and the guy behind the company’s SuperBowl ads. Judge blogs and has a 22,000 following on Twitter. There’s a guy who understands Best Buy customers.

Know the future.

It’s worth looking for a new CEO with an ability to see change coming—and to respond to it before it happens. As contributor Larry Downes pointed out in his widely-read post, Best Buy has made maddening errors that it should have avoided, like the Christmas merchandise shortage. It’s not like the Internet is a secret. Best Buy should be one of the most responsive, customer-friendly e-commerce sites out there. The new CEO needs to be someone who gets this done, rather than blaming the products or customers.

What about a left-field choice, like someone from a consumer electronics company such as Samsung, Dell or HP?

"Brian Dunn has said in the past that 50% of our hard goods are going to be made by us," Goldberg says. "If the board believes that, you might look at a senior exec from Sony, or a younger consumer electronics company." Vizio, a flat-screen TV maker, or HTC might yield some execs, Goldberg said.